My research interests lie in the areas of applied harmonic analysis, applied linear algebra, and approximation theory, with special focus on frame theory and signal processing. Publications

A key idea in harmonic analysis is to decompose a given function into "simpler" components that make it easier to study the properties of the function. When dealing with finite dimensional spaces, most things can be done using linear algebra. In infinite dimensions, one needs tools from functional analysis and things are more subtle. For example, one might have to take into account the various notions of basis and/or deal with issues arising from convergence of infinite sums. Below is a brief description of some of the things I study that might interest students who have taken some undergraduate linear algebra course, and are familiar with the concept of a basis and its fundamental properties. A nominal knowledge of complex numbers is needed.

Finite Frame Theory (Applied Linear Algebra): In linear algebra, one of the basic ideas is to express a given object in some space in terms of elements in a representative set like a basis. When dealing with different kinds of data sets, the structure of this representative set becomes crucial for efficient storage and transmission of data. Frames are representative sets like bases but are redundant. The redundancy allows more flexibility and freedom of choice. Frames have now become standard tools in signal processing due to their resilience to noise and transmission losses.

Project ideas appropritae for undergraduate students

• Study and compare the effect of different kinds of frames in signal (image) reconstruction. Certain frames turn out to be better than others. Determine properties or characteristics of frames that perform better.
• Frame design: In some situations, one might seek a "sparse" representation of a signal. In other situations, one might have to use a subset of a frame to approximate a signal or to deal with loss. Given the goal, properties like "equiangularity", "equal-norm", or "tightness" might be desirable. Consequently, one wishes to construct frames having some "desirable" properties.
• Frame transformations: Starting with a frame, investigate the action of certain "transformations" on the given frame. Which properties of the starting frame are preserved? Determine transformations that can add some property missing in the original frame.
Here is a book chapter on undergraduate research in Finite Frame Theory that appears in A Primer for Undergraduate Research: From Groups and Tiles to Frames and Vaccines of the series Foundations for Undergraduate Research in Mathematics, Springer, 2017.

When working with infinite dimemsional function spaces, generalizing some concepts such as equiangularity becomes tricky, but one might seek frames with a specific structure such as a Gabor structure or a wavelet structure or even frames coming from sets of orthogonal polynomials. Such studies involve concepts from real and functional analysis. Another interesting study is that of frames or frame properties of sets that come from the action of elements of a group on a single vector. If you are a graduate student interested in working in the area of applied harmonic analysis or frame theory feel free to send me an email.